(Redirected from Unit)
There are no approved revisions of this page, so it may
not have been
reviewed.
This article page is a stub, please help by expanding it.
An element of a set is a unit if its multiplicative inverse belongs to that set. (The set of units form a multiplicative group.) Algebraic integer units have complex norm 1 and are thus complex roots of unity.
Units in quadratic integer rings
Imaginary quadratic integer rings have only a few units (see Theorem I1). The primitive root of unity of degree 3
 ${\begin{array}{l}\displaystyle {\omega :=e^{i{\tfrac {2\pi }{3}}}={\frac {1}{2}}+{\frac {\sqrt {3}}{2}},}\end{array}}$
is used in the following table.
Units of imaginary quadratic integer rings

Units

≤ − 5

{( − 1) 0, ( − 1) 1} = {1, − 1} 

− 3

{ω 0, ω 1, ω 2, ω 3, ω 4, ω 5} = {1, ω, ω 2, − 1, − ω, − ω 2} 

− 2

{( − 1) 0, ( − 1) 1} = {1, − 1} 

− 1

{i 0, i 1, i 2, i 3} = {1, i, − 1, − i} 

Real quadratic integer rings have infinitely many units (see Theorem R1), which are all powers of each other, some multiplied by 1. For that reason, the table below can’t be complete like the table above.
An inefficient way to find a unit of a real quadratic integer ring
other than
1 or
1 is to try each positive value of
starting with
1 and going up until
is an integer.
Units of real quadratic integer rings

Units

2

$1+{\sqrt {2}}$

3

$2+{\sqrt {3}}$

4

N/A

5

${\frac {1}{2}}+{\frac {\sqrt {5}}{2}}$ (golden ratio)

6

$5+2{\sqrt {6}}$

7

$8+3{\sqrt {7}}$

8

N/A, but note that $3^{2}8(1^{2})=1$

9

N/A

10

$3+{\sqrt {10}}$

11

$10+3{\sqrt {11}}$

12

N/A, but $7^{2}12(2^{2})=1$

13

${\frac {3}{2}}+{\frac {\sqrt {13}}{2}}$

14

$15+4{\sqrt {14}}$

15

$4+{\sqrt {15}}$

16

N/A

17

$4+{\sqrt {17}}$

18

N/A, but $17^{2}18(4^{2})=1$

19

$170+39{\sqrt {19}}$

20

N/A, but $9^{2}20(2^{2})=1$

21

${\frac {5}{2}}+{\frac {\sqrt {21}}{2}}$

22

$197+42{\sqrt {22}}$

23

$24+5{\sqrt {23}}$

24

N/A, but $5^{2}24(1^{2})=1$

25

N/A

26

$5+{\sqrt {26}}$

27

N/A, but $26^{2}27(5^{2})=1$

28

N/A, but $127^{2}28(24^{2})=1$

29

${\frac {5}{2}}+{\frac {\sqrt {29}}{2}}$

30

$11+2{\sqrt {30}}$

31

$1520+273{\sqrt {31}}$

32

N/A, but $17^{2}32(3^{2})=1$

33

${\frac {23}{2}}+{\frac {4{\sqrt {33}}}{2}}$

Examples
See also